# Toronto Math Forum

## MAT244--2018F => MAT244--Tests => Quiz-7 => Topic started by: Victor Ivrii on November 30, 2018, 04:12:23 PM

Title: Q7 TUT 5102
Post by: Victor Ivrii on November 30, 2018, 04:12:23 PM
(a) Determine all critical points of the given system of equations.

(b) Find the corresponding linear system near each critical point.

(c) Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system?

(d)  Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system.
\left\{\begin{aligned} &\frac{dx}{dt} = (2 + x)( y - x),\\ &\frac{dy}{dt} = (4 - x)( y + x). \end{aligned}\right.

Bonus: Computer generated picture
Title: Re: Q7 TUT 5102
Post by: Chonghan Ma on November 30, 2018, 04:19:46 PM
(a)
Set (2+x)(y-x)=0 and (4-x)(y+x)=0
Then we have critical points (0,0), (4,4), (-2,2)
(b)
J = \begin{bmatrix}-2-2x+y & 2+x \\4-2y-2x & 4-x \end{bmatrix}
Linear systems are shown with each critical point:
J(0,0) =  \begin{bmatrix}-2 & 2 \\4 & 4 \end{bmatrix}
J(-2,2) =  \begin{bmatrix}4 & 0 \\6 & 6 \end{bmatrix}
J(4,4) =  \begin{bmatrix}-6 & 6 \\-8 & 0 \end{bmatrix}
(c)
Eigenvalues are computed by det(A - tI)= 0
So that
At (0,0): t=1±√17}
Critical point is a saddle point and it is unstable
At (-2,2): t= 4 and 6
Critical point is an unstable node
At ((4,4): t=-3±√9 i
Critical point is a stable spiral point
Title: Re: Q7 TUT 5102
Post by: Xiaoyuan Wang on November 30, 2018, 04:43:46 PM